Assuming your table stores the start and end dates in 2 seperate columns then you simply need to make sign of the difference in dates for any row is the same.
Ie
row Start End
1 1/14 1/16
2 1/21 1/25
New request 1/15  1/17
So you run datediff (or whatever function in access/excel)
1/14 and 1/15 = 1
1/16 and 1/15 = 1 the signs are different so 1/15 is between 1/14 and 1/16
request 1/18  1/19
1/14 and 1/18 = 4
1/16 and 1/18 = 2 then signs are same so 1/18 is outside of 1/14 and 1/16
1/14 and 1/19 = 5
1/16 and 1/19 = 3
As long as the sign is the same any start/end pair then the date you are checking is not in the range.
Do the same for all the previous bookings and you're all set.
